Connectedness of Hilbert function strata and other connectedness results

The main purpose of this paper is to give a proof that any Hilbert function stratum, i.e., the set of the points of a Hilbert scheme with a fixed Hilbert function, is connected in characteristic zero. Furthermore, we give a criterion when the union of two (or more) Hilbert function strata is itself connected. We also give short proofs of the theorems of Gotzmann and Hartshorne in characteristic zero. Furthermore, we prove the connectedness of the subsets of the Hilbert scheme consisting of points with a fixed Castelnuovo–Mumford regularity, as well as the connectedness of the intersections of these sets with Hilbert function strata.